In a lot of fiction, there's the image of the bandit: roving brigands, one specific tough leader, who set up camps with tents and demand your money or your life. How common was this profession in the middle ages?

Depends on how you mean by 'common'. Banditry pops up every so often here, and these three excellent answers should cover most of your needs:

u/idjet has an answer in this thread and also links back to one of their previous answers on the same topic;
u/Wagrid examines two particular bandit gangs in this thread;
and u/sunagainstgold also has an answer in this thread.